Angles

In standard position, the initial side lies on the positive x-axis. Moving anticlockwise, \(90^\circ\) reaches the positive y-axis and \(180^\circ\) the negative x-axis; hence \(270^\circ\) lies on the negative y-axis. Exam tip: remember the axis order every \(90^\circ\).

(90^\circ) is half of (180^\circ) so it is ( \frac{\pi}{2} ). Divide the degree measure by (180^\circ) first.

Coterminal angles have the same initial and terminal sides, and their difference is an integral multiple of \(360^\circ\). Since \(30^\circ+360^\circ=390^\circ\), A is correct. The difference for \(330^\circ\) is \(300^\circ\), not \(360^\circ\). Exam tip: use \(\theta\pm360^\circ\).

(30^\circ=\frac{30\pi}{180}=\frac{\pi}{6}). Multiply degrees by ( \frac{\pi}{180} ) to convert to radians.

( \pi ) radians equals (180^\circ). Multiply by ( \frac{180^\circ}{\pi} ) to convert radians to degrees.

To convert radians into degrees, multiply by \(\frac{180^\circ}{\pi}\): \(\frac{\pi}{3}\times\frac{180^\circ}{\pi}=60^\circ\). Hence, \(60^\circ\) is correct. \(30^\circ\) is a close distractor because its radian measure is \(\frac{\pi}{6}\), not \(\frac{\pi}{3}\). Exam tip: Remember that \(\pi\) radians = \(180^\circ\).

( \frac{\pi}{6}\times \frac{180^\circ}{\pi}=30^\circ). For radians to degrees always use ( \frac{180^\circ}{\pi} ).

( \frac{\pi}{4} ) radians is (45^\circ). Use ( \pi ) radians as (180^\circ) and form a ratio.

A positive angle is measured in the anticlockwise direction. In sign-based questions always check the direction.

In standard position, an angle is measured from its initial arm. Angles measured anticlockwise are positive, whereas angles measured clockwise are negative. Therefore, the correct answer is clockwise direction. Anticlockwise is the closest distractor because it represents positive angles. Exam tip: remember the sign convention—anticlockwise (+), clockwise (−).

One complete revolution is (360^\circ). Adding or subtracting (360^\circ) helps in coterminal angles.

A straight angle forms a straight line, so its measure is \(180^\circ\). An angle of \(90^\circ\) is a right angle, while \(360^\circ\) is a complete angle. Exam tip: whenever an angle forms a straight line, use \(180^\circ\).

A right angle is (90^\circ) and its radian measure is ( \frac{\pi}{2} ). Remember standard conversions.

(360^\circ) is two times (180^\circ) so it is (2\pi) radians. Remember the radian measure of a full turn.

(270^\circ=\frac{270\pi}{180}=\frac{3\pi}{2}). Quickly identify multiples of (90^\circ).

To convert degrees to radians, multiply the angle by \(\frac{\pi}{180}\). Thus, \(120^\circ=120\times\frac{\pi}{180}=\frac{2\pi}{3}\) radians. Note that \(\frac{5\pi}{6}\) radians equals \(150^\circ\), so it is not correct here. Exam tip: Remember that \(180^\circ=\pi\) radians.

(210^\circ=\frac{210\pi}{180}=\frac{7\pi}{6}). Multiply degrees by ( \frac{\pi}{180} ).

Angles in the third quadrant are greater than \(180^\circ\) and less than \(270^\circ\). Since \(210^\circ\) lies in this interval, it is correct. \(135^\circ\) is in quadrant II. Exam tip: check the interval first.

(300^\circ=\frac{300\pi}{180}=\frac{5\pi}{3}). Recognizing multiples of (60^\circ) is helpful.

\(45^\circ\) is greater than \(0^\circ\) and less than \(90^\circ\), so it is an acute angle. \(90^\circ\) is a right angle and is not in this open interval. Exam tip: check whether endpoint values are included.